Until today we have dealt with internal angles, perhaps also with adjacent angles, but we have not talked about external angles. Don't worry, the topic of the exterior angle of a triangle is very easy to understand and its property can be very useful for solving exercises more quickly. Shall we start?
What is the exterior angle of a triangle?
The exterior angle of a triangle is the one that is found between the original side and the extension of the side.
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Imagine someone draws a triangle and falls asleep as they are finishing it. Without realizing it, they continue drawing one side a little more... and Bam! An exterior angle is created. The exterior angle is outside of the triangle and is found between the original side and the side they continued drawing while asleep (the continuation of the side).
Let's look at an example
Observe: An exterior angle is the one that is found between an original side of the triangle and the extension of the side and not between two extensions.
Note: Whenever the angle is outside the triangle and is found between an original side of the triangle and the extension of another side of the triangle, it will be considered an exterior angle of the triangle.
Great! Now that we have understood what an exterior angle is and that we can recognize it from a distance, we can move on to the property of the exterior angle of a triangle. Property of the exterior angle of a triangle The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.
Given that: ∢A=80 ∢B=20
How much does the exterior angle measure? Solution: Let's denote the exterior angle withα:
According to the property of the exterior angle of the triangle, the exterior angle α must be equal to the sum of the two interior angles of the triangle that are not adjacent to it. That is, ∢A+∢B
We already have these angles. Therefore, all we have to do is add them up and find out the exterior angle: α=80+20 α=100
Look, we could have found the value of the exterior angle in another way! We know that the sum of the interior angles of a triangle is 180. Therefore, ∢ACB=180−20−80
∢ACB=80
∢ABC is the angle adjacent to α, the exterior angle of the triangle that we need to find out. We also know that the sum of the adjacent angles is 180. Therefore we can determine that: ∢80+α=80 α=100
Look, In certain cases you will not be explicitly asked for the value of the exterior angle. They might ask you, for example, about some interior angle of the triangle that you could figure out through the exterior angle.
Let's look at an example
Given the following triangle:
Data: ∢A=90 α=110
Find the value of ∢B
Solution:
We can solve the problem in two ways:
The first is based on the Exterior Angle Theorem of a triangle and understand that α is an exterior angle of the triangle and is equal to the sum of the two interior angles that are not adjacent to it. That is, ∢A+∢B
Then, the equation would be: 110=90+∢B ∢B=20
The second way to solve the problem is to remember that the sum of the adjacent angles equals 180, then ∢ACB is equal to 70.
Notice that we have arrived at the same result, but solving through the property of the exterior angle of a triangle has been faster to reach it.
Useful Information: The sum of the three exterior angles of a triangle equals 360 degrees.
In conclusion, it is important and really worth knowing the property of the exterior angle of a triangle to solve problems easily and quickly, although in several cases you will be able to manage without this magnificent theorem.
Examples and exercises with solutions of an exterior angle of a triangle
Exercise #1
In a right triangle, the sum of the two non-right angles is...?
Video Solution
Step-by-Step Solution
In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)
Therefore, the sum of the two non-right angles is 90 degrees
90+90=180
Answer
90 degrees
Exercise #2
Can a triangle have two right angles?
Video Solution
Step-by-Step Solution
The sum of angles in a triangle is 180 degrees. Since two angles of 90 degrees equal 180, a triangle can never have two right angles.
Answer
No
Exercise #3
Look at the two triangles below. Is EC a side of one of the triangles?
Video Solution
Step-by-Step Solution
Every triangle has 3 sides. First let's go over the triangle on the left side:
Its sides are: AB, BC, and CA.
This means that in this triangle, side EC does not exist.
Let's then look at the triangle on the right side:
Its sides are: ED, EF, and FD.
This means that in this triangle, side EC also does not exist.
Therefore, EC is not a side in either of the triangles.
Answer
No
Exercise #4
Calculate the size of angle X given that the triangle is equilateral.
Video Solution
Step-by-Step Solution
Remember that the sum of angles in a triangle is equal to 180.
In an equilateral triangle, all sides and all angles are equal to each other.
Therefore, we will calculate as follows:
x+x+x=180
3x=180
We divide both sides by 3:
x=60
Answer
60
Exercise #5
Calculate the size of the unmarked angle:
Video Solution
Step-by-Step Solution
The unmarked angle is adjacent to an angle of 160 degrees.
Remember: the sum of adjacent angles is 180 degrees.
Therefore, the size of the unknown angle is:
180−160=20
Answer
20
Check your understanding
Question 1
In an isosceles triangle, the third side is called?